bosonic fractionalisation transitions
TRANSCRIPT
Imperial/TP/2012/JS/02
Bosonic Fractionalisation Transitions
Alexander Adam1), Benedict Crampton1), Julian Sonner1),2), Benjamin Withers3)
1)Theoretical Physics Group, Blackett Laboratory,Imperial College, Prince Consort Rd, London SW7 2AZ, U.K.
2)DAMTP, University of Cambridge,C.M.S. Wilberforce Road, Cambridge, CB3 0WA, U.K.
3)Centre for Particle Theory & Department of Mathematical Sciences,
Science Laboratories, South Road, Durham DH1 3LE, U.K.
Abstract
At finite density, charge in holographic systems can be sourced either by
explicit matter sources in the bulk or by bulk horizons. In this paper
we find bosonic solutions of both types, breaking a global U(1) symmetry
in the former case and leaving it unbroken in the latter. Using a minimal
bottom-up model we exhibit phase transitions between the two cases, under
the influence of a relevant operator in the dual field theory. We also embed
solutions and transitions of this type in M-theory, where, holding the theory
at constant chemical potential, the cohesive phase is connected to a neutral
phase of Schrodinger type via a z = 2 QCP .
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1 Introduction
Given a holographic system at finite density with respect to some U(1) symmetry, we
would like to answer basic questions relating to the physical nature of its groundstate
such as ‘is the symmetry broken or unbroken?’ and in the latter case ‘are there Fermi
surfaces present?’ and ‘what are the properties of such Fermi surfaces?’. Lately such
questions have attracted a great deal of interest leading to general statements along
the lines of a holographic Luttinger relation and its bosonic analogue [1, 2, 3]. The
Luttinger count relates the sum of the volumes of all Fermi surfaces present to the total
charge, while its bosonic analogue relates the total charge to the ‘Magnus force’ felt
by a test vortex in the dual field theory. Such studies were guided by the realisation
that the finite charge density of a field theory - as encoded in the dual geometry
via a finite electric flux - can be sourced either by explicit charged matter, such as
fermions or charged bosonic condensates, or by a charged horizon. The latter situation
corresponds to having some of the charge density of the field theory tied up in gauge-
variant operators, and correspondingly not contributing to a naively defined Luttinger
count or its bosonic analogue. The charge contributing to this deficit is identified with
so-called ‘fractionalised’ charge in the field theory, that is charge carried by deconfined
‘quark’ degrees of freedom which are charged under the gauge symmetry. This point
of view has recently been bolstered by the identification of the corresponding 2kF
singularities associated with the ‘hidden’ fermionic degrees of freedom by Faulkner and
Iqbal [4].
There can be interesting phase transitions between situations in which flux is sourced
by a horizon to ones where it originates from explicit matter sources in the bulk [1, 5, 6].
An early example of such a transition was observed in the M-theory superconductor [7,
8], where it was demonstrated that the zero-temperature limit of the superconducting
black hole was a charged domain-wall geometry without horizon due to the interplay
of the potential and effective gauge coupling as defined by a neutral scalar field.
The ground states of the Abelian Higgs model with a W-shaped potential were
analysed in [9, 10], where it was observed that the nature of the ground state can
change qualitatively as a function of the gauge coupling. In [7, 8] it was also pointed
out that the disappearance of the horizon as T → 0 can be related to Nernst’s Law since
the charged domain wall has zero entropy, whereas the naive extrapolation of the finite-
temperature geometry to an extremal black hole would carry entropy. It is thus perhaps
expected that the charged domain wall is thermodynamically preferred over the finite-
1
entropy black hole solutions in precisely that region of the phase diagram where both
solutions coexist and thus the ‘entropic singularity’ of the M-theory superconductor is
cloaked by a superconducting dome.
In this work we explore bosonic analogues of the fractionalisation transition of [1],
corresponding to transitions from phases with broken U(1) symmetry to phases with
intact U(1) symmetry and all charge associated with a horizon. The former consti-
tutes the analogue of the ‘mesonic phase1’ of [1] and the latter corresponds to a fully
fractionalised phase. There may also be partially fractionalised phases, in which some
of the charge is carried by the horizon and some by the scalar matter. Because of
the presence of non-singlet matter, the partially fractionalised phase always breaks the
U(1) symmetry, so that the transition between a cohesive and a partially fraction-
alised phase must occur within the superfluid state. Hence we refer to the two kinds
of ordered phases in this paper as the ‘superfluid cohesive phase’ and the ‘superfluid
fractionalised phase’. The transition between the cohesive and fractionalised phase has
striking implications for the full phase diagram (see Fig. 3), yet at this point no field-
theoretical order parameter is known for this transition. We study these transitions in
a variety of models, both in a ‘bottom-up’ and a ‘top-down’ context, with a range of
different behaviours, and we describe the conditions under which fractionalisation tran-
sition can occur. Among other things we describe a supergravity-embedded superfluid
cohesive to neutral phase transition, which has certain aspects that are reminiscent of
the standard superfluid to insulator transition in the Bose-Hubbard model at constant
chemical potential.
It is worth commenting that the theories considered in this paper have a structural
similarity to the Thomas-Fermi approach used in the electron-star literature [12] with
the charged scalar field taking on the role of the fluid of bulk fermions. It should not
come as a surprise then that some of the IR geometries encountered in those studies
will also feature in the bosonic theories here.
An important role is played in the present work by IR geometries with certain scaling
or hyperscaling properties. Such solutions have previously found applications in applied
holography, for example in describing QCD like theories [13, 14, 15], and more recently
in describing quantum-critical matter at finite density [16, 17, 18, 19, 20, 21, 22, 23].
The paper is organised as follows. Immediately following this paragraph we briefly
summarise our results. In section 2 we introduce the class of Einstein-dilaton-charged
1Apart from this one instance we adopt the term ‘cohesive’ for any non-fractionalised phase [11],in order to avoid potential confusions with the more conventional meaning of the term ‘mesonic’.
2
scalar models used throughout the paper and develop their thermodynamics and holo-
graphic renormalisation. Section 3 gives a detailed discussion of certain bottom-up
models which illustrate the general concept of bosonic fractionalisation and motivate
the more subtle M-theory case to which we turn in section 4. We finish with a discussion
and a short appendix.
1.1 Summary of results
The salient features of the analysis boil down to the IR structure of the zero-temperature
states. We encounter three cases, as illustrated in Fig. 1, depending on the structure
of the effective gauge coupling, which is controlled by the neutral scalar field. If the
gauge coupling vanishes in the IR, we find a fully fractionalised phase, or a superfluid
fractionalised phase. If it remains finite, we find a cohesive phase. The U(1) symmetry
is broken in the latter two phases by explicit charged matter outside the horizon. In ad-
dition to the fully fractionalised phase, there is one further unbroken phase, where the
neutral groundstate does not depend on the chemical potential. This ‘incompressible’
phase together with the transition into it from the cohesive phase via a z = 2 criti-
cal point is reminiscent of the superfluid to insulator transition in the Bose-Hubbard
model.
2 General Features
In this section we introduce the theories leading to quantum phase transitions in terms
of a framework that is general enough to cover both the bottom-up constructions of
section 3 and the M-theory compactifications of section 4. Our general strategy follows
a two-pronged approach. Firstly, at zero temperature we will carry out an analysis of
possible IR geometries and the deformations which allow them to be connected to the
asymptotic AdS configuration universal to all solutions considered in this paper. We
are interested both in IR geometries associated with domain wall configurations and IR
geometries associated with (extremal) black-hole horizons. This allows us to classify
the possible ground states of the holographic theories in this work. Secondly we will
extrapolate the known finite-temperature solutions and their dilatonic deformations
to very low temperatures. These can lead to a state with broken U(1) symmetry via
a condensation instability at some critical temperature Tc or to a state in which this
symmetry is intact. In each case we are able to connect them to the appropriate T = 0
3
class Ia
class Ib
class II
superfluid cohesive
gc gf
superfluid f’ised fully f’ised
neutral superfluid cohesive neutral
fully f’ised superfluid f’ised fully f’ised
gC gC
gf gfgc
Figure 1: Zero-temperature phase structure of the the models considered in thispaper. Shades of red indicate broken U(1) symmetry, with dark red symbolisingthe cohesive states and bright red the superfluid fractionalised ones. The U(1)symmetry is unbroken in the green fully fractionalised and grey neutral phases.The class Ib example is even in Φ, while the class Ia example has no partic-ular discrete symmetry associated with the dilaton. Note that the M-theoryembedding in section 4 is in class II.
geometry. In this way we can also study other interesting low-temperature signatures,
such as the dynamical critical exponent z and the hyperscaling violation exponent θ,
in terms of which the low-temperature entropy scales like
S ∼ Td−θz . (2.1)
We are able to identify z and θ from an appropriate (hyper-)scaling IR geometry,
matching perfectly to numerical fits of our low-temperature analysis.
2.1 Action and Field Content
The class of theories we work with in this paper is described by an action of the form
S =
∫dd+1x
√−g[R− 1
4ZF (Φ)FMNF
MN − 12(∇Φ)2 − ZS(S)|DS|2 − V (Φ, |S|)
]+ SCS .
(2.2)
4
We denote the d+1 bulk directions with upper-case Latin M,N, . . ., whilst field-theory
coordinates will be denoted with lower-case Greek µ, ν, . . .. The field Φ is a neutral
scalar field, which we will refer to as a ‘dilaton’, whereas S is a charged scalar field
with covariant derivative DS = (∇− iqA)S. We have included SCS, which stands for a
possible Chern-Simons or axion-like contribution often found in consistent supergravity
reductions. Whilst the solutions we construct do not depend on SCS, we note that in
certain cases the phase structure may be altered by its presence, for example through
the spontaneous formation of stripes.
We allow the couplings of the gauge field, as well as the charged scalar kinetic
terms, to depend on Φ and S respectively. The interesting transitions observed in the
remainder of this paper can be traced back to the interplay of charged and neutral
scalar couplings and the potential. Often it is convenient to put the scalar kinetic
terms into canonical form. A little thought shows that this is only possible for the
magnitude of the complex field. Writing S in a polar decomposition with phase iqϕ
and then using a field redefinition of the magnitude, we can write the kinetic term of
the charged scalar as
ZS(|S|)|DS|2 = (∇η)2 + q2X(η)2(∇ϕ− A)2 . (2.3)
We should require that X(η) has a small-η expansion X(η) ∼ η +O(η2). The hyper-
scaling geometries are generated by a divergent IR dilaton Φ −→ ±∞, and so it is
important to distinguish theories where such a divergence can happen from those were
it cannot:
class I : ZF (Φ)IR−−−−−→ ∞ ,
class II : ZF (Φ)IR−−−−−→ (always stays bounded) . (2.4)
Furthermore ZF (Φ) may or may not have even symmetry under Φ → −Φ as the
IR limit is approached, again with important implications for the phase structure at
zero temperature. In class I we are particularly interested in ZF (Φ) without this
symmetry, so that e.g. if ZF (Φ) diverges as Φ −→ +∞ then it remains bounded when
Φ diverges with the opposite sign. In this manner the dilaton can naturally drive a
fractionalisation-type phase transition.
Finally we require that the potential has an expansion
`2V (η,Φ) = −6− Φ2 − 2η2 + · · · , (2.5)
5
ensuring that there exists an AdS4 vacuum solution with associated AdS length `2,
and that the two scalars2 Φ and S are dual to ∆ = 2 operators. For the solutions in
this paper we can take the scalar S identically real, so that without loss of generality
ϕ ≡ 0.
Throughout this paper we use the metric ansatz
ds2 = −f(r)e−β(r)dt2 +dr2
f(r)+r2
`2
(dx2 + dy2
). (2.6)
2.2 Holographic renormalisation and asymptotic charges
We are interested in solutions that asymptotically approach the AdS4 fixed point in
the UV. Such solutions admit a large r expansion
f =r2
`2+
1
2
(η2
1 + 12Φ2
1
)+`G1
r+ · · · ,
β = βa +`2(η2
1 + 12Φ2
1
)2r2
+ · · · ,
Φ =`Φ1
r+`2Φ2
r2+ · · · ,
η =`η1
r+`2η2
r2+ · · · ,
At = `e−βa/2(µ− Q
r+ · · ·
). (2.7)
By evaluating the asymptotic stress tensor (which involves introducing counterterms
to render the renormalised expressions finite) we can deduce the exact relationship of
the integration constants introduced here and the physical charges of the dual field
theory. This is achieved by adding the specific counterterm action
Sct = 2
∫∂Σ
√−γ(K − 2
`
)−∫∂Σ
√−γ 1
`
(|S|2 + 1
2Φ2), (2.8)
where we have chosen the counterterms for the scalar fields such that the fixed η1,Φ1
ensemble has a well defined variational principle. K = γµνKµν is the trace of the
extrinsic curvature of a constant r hypersurface with unit normal nr = f(r)12 . We find
that the total action
Sren = S + Sct (2.9)
2The seemingly different mass terms are caused by the different normalisations of the respectivekinetic terms. We are using a convention that appears to be standard in the literature.
6
is finite, as is the renormalised stress tensor
Tµν = 2
[Kµν −Kγµν −
2
`γµν
]− 1
`
(|S|2 + 1
2Φ2)γµν . (2.10)
From this we can relate the expansion coefficient G1 to the energy density3, defined as
ε =reβa
`T00 = −2
`
(G1 − η1η2 − 1
2Φ1Φ2
). (2.11)
The two scalar fields play very different roles in our system. The dilaton Φ gives rise
to an explicit deformation parameter: we switch on a relevant operator by sourcing it
in the dual theory, i.e. by taking a non-zero value of the UV parameter Φ1, and then
study the behaviour of the theory as we vary this source. In contrast, the field η is
only ever allowed to condense spontaneously, i.e. we set η1 = 0.
2.3 Thermodynamic Relations
By analytically continuing to Euclidean signature
t = −iτ , IE = −iSren , (2.12)
we define the grand canonical potential
IE = βΩ(µ, T ) = βvol2ω(µ, T ) . (2.13)
Evaluated on-shell, the action can be written as a total derivative
SOS = iβvol2
∫ ∞r+
d
dr
[2
r
√−ggrr
]dr , (2.14)
which vanishes at the horizon, and thus depends only on the asymptotic charges. Using
techniques of [24, 25] one can show that this is the negative of the pressure
ω(µ, T ) = −P . (2.15)
We can exploit the symmetries of the background metric to derive a Smarr-Gibbs-
Duhem relation. This follows from the vanishing of a certain total derivative4 on shell,
3The quantity e−βa is the boundary speed of light (squared), and so the conversion between ADMmass and energy, and correspondingly energy density, involves a factor of e−βa . For this and relatedreasons it is convenient to set βa = 0, which we will assume to be the case from now on.
4The on-shell vanishing of the total derivative explains why the authors of [8, 24] were able to writethe on-shell action as two different boundary terms.
7
as we now show. By using the Killing equation in addition to the Ricci identity for the
two Killing vectors ∂t and ∂y we can derive the relation
√−g(Rtt −Ry
y
)= −∂r
[e−β/2r2
2`2
(fβ′ − f ′ + 2f
r
)]. (2.16)
It is then a simple matter of using the trace-removed Einstein equations to find that
on-shell√−g(Rtt −Ry
y
)= −1
2∂r[√−gZ(φ)AtF
tr], (2.17)
so that, upon integrating from the horizon to the UV boundary, we obtain the relation
e−β/2r2
`2
(fβ′ − f ′ + 2f
r
)∣∣∣∣∣∞
r+
=√−gZ(φ)AtF
tr∣∣∣∞ . (2.18)
Evaluated on our expansions this gives the Gibbs-Duhem relation
3
2ε = µQ+ Ts− `−1
(η1η2 +
1
2Φ1Φ2
), (2.19)
or, written in terms of the pressure
ε+ P = µQ+ Ts . (2.20)
Here we have used that the temperature is given by
T =e(βa−β+)/2
4π`2f ′(r)
∣∣∣r=r+
, (βa = 0) . (2.21)
Moreover, the entropy density is given in terms of the horizon radius as
s =r2
+
4GN
, (note 16πGN = 1) . (2.22)
The above derivations continue to make sense for solutions of the theory (2.2) where
limr→rIR f′(r) = 0 as rIR → 0, i.e. solutions of vanishing entropy at zero temperature.
We also allow the dilaton field to diverge logarithmically5 in this limit, as long as it
does not interfere with the conditions just mentioned.
5Specifically the dilaton blows up logarithmically as a function of radial proper distance.
8
3 Class I : A Bottom-Up Model
When Z is not symmetric in Φ there is the possibility of two qualitatively different
behaviours assosciated with a dilaton that diverges logarithmically to either plus or
minus infinity. In the following models, we will see that in the first case the effective
gauge coupling e2 ∼ Z−1 will go to zero in the IR and in the second it will diverge. There
is a continuous transition between a superfluid fractionalised phase and a cohesive
phase, via a quantum critical point with a finite dilaton. Both phases will be of
hyperscaling form. This phase transition will be driven by the asymptotic value of the
dilaton. We will also see an additional phase transition between a partially and a fully
fractionalised phase, assosciated with the edge of the superconducting dome.
We will consider the class of models
ZF (Φ) = Z20eaΦ/√
3 , `2V (Φ, η) = −V 20 cosh
(bΦ/√
3)− 2η2 + g2
ηη4 , (3.1)
where a, b > 0. The η4 term allows for interpolating soliton solutions between the AdS4
maximum at η = 0 and the AdS4 minimum at η2 = g−2η . We will study IR geometries
for general values of a and b (see also [1, 11]). All our numerical results will be for the
particular case a = b = g2η = 1, V 2
0 = 6 and Z20 = 1.
3.1 Zero Temperature
We now study the phases of our system at zero temperature. We construct the IR
geometries by making a scaling ansatz for the fields, allowing for a possible logarithmic
divergence of the dilaton. With this divergence the potential is schematically V ∼rδ + r−δ, for some δ, and so these scaling solutions are not exact: they are the leading
order parts of series solutions, with subleading corrections in increasing powers of r.
For each of our three cases we also identify the possible irrelevant deformations.
The divergent dilaton endows our leading order solutions with hyperscaling symme-
try [22, 20, 21], which is a generalisation of the familiar Lifshitz scaling:
x → λx ,
t → λzt ,
r → λ(θ−2)/2r , (3.2)
under which ds → λθ/2ds. Here z is the usual dynamical critical exponent, and θ is
the hyperscaling violation parameter.
9
3.1.1 Fractionalised IR Solutions
In this case some or all of the flux is sourced by the horizon. Starting with the fully
fractionalised case, we switch off the condensate. We find the following scaling solutions
at leading order:
f(r)e−β(r) = r2(12+a2−b2)
(a+b)2 ,
f(r) = f0r2(a−b)a+b ,
At(r) = A0r12+(3a−b)(a+b)
(a+b)2 ,
Φ(r) = − 4√
3
a+ blog r , (3.3)
with
f0 =(a+ b)4V 2
0
4(6 + a(a+ b))(12 + (3a− b)(a+ b)),
A20 =
4(6− b(a+ b))
(12 + (3a− b)(a+ b))Z20
. (3.4)
These solutions have dynamical critical exponent z and hyperscaling parameter θ given
by
z =12 + (a− 3b)(a+ b)
a2 − b2, θ =
4b
b− a. (3.5)
Note that though these parameters are infinite in our special case a = b = 1, their
ratio is still finite (such a situation is discussed further in [11]). This ensures that for
solutions close to this T = 0 geometry, the various thermodynamic parameters have
finite scaling exponents with temperature.
There are two irrelevant deformations about this solution. The first simply corre-
sponds to a shift in the IR vacuum expectation value of the charged scalar:
δη = η0 . (3.6)
Integrating this deformation to the UV gives geometries with explicit bulk charge in
addition to horizon charge. We thus find that the fully fractionalised solutions connect
continuously to the superfluid fractionalised phase branch of solutions. The second
deformation is assosciated with the dilaton, and has the form
δΦ = aIRΦ rνΦ , δf = aIRf rνf , δβ = aIRβ rνΦ , δAt = aIRA rνA , (3.7)
10
with exponents
νf = νΦ +2(a− b)a+ b
,
νA = νΦ +12 + (3a− b)(a+ b)
(a+ b)2, (3.8)
where the coefficients aI and the exponent νΦ are consistently determined by the IR
limit of the linearised equations of motion. For example, in the simplified case a = b
the theory admits an irrelevant deformation with exponent
νΦ =3 + b2
6b4
(−3 +
√81− 24b2
), (3.9)
and coefficients
af , aβ , aA , aΦ = aIR−f0 , 2 ,
A0νΦb2
2(3− b2),− b√
3
, (3.10)
with an overall free magnitude aIR. This is to be compared with the analogous mode
in [1].
3.1.2 Critical IR Solution
The critical IR solution has non-divergent dilaton (in fact Φ = 0) and is just the AdS4
minimum that arises when the condensate sits at the bottom of its potential, η = ±g−1η ,
with vanishing gauge field At(r) = 0. Consequently its AdS length scale is shifted away
from the usual value by a gη-dependent factor:
f(r)e−β(r) = r2 ,
f(r) =r2
R2IR
,
η(r) = ±g−1η , (3.11)
with
R2IR =
6g2η`
2
1 + g2ηV
20
. (3.12)
There are two irrelevant deformations about this solution: one involving the flux and
one the charged condensate. They have the form:
δAt = aIRA rνA , δη = aIRη rνη , (3.13)
11
with exponents
νA =1
2
(−1 +
√1 +
48`2q2
Z20
(1 + g2
ηV2
0
)) ,
νη =3
2
(−1 +
√1 +
32g2η
3(1 + g2
ηV2
0
)) . (3.14)
3.1.3 Cohesive IR Solution
We seek cohesive solutions by demanding that the horizon flux limr→0
√−gZ(Φ)F rt
vanishes. At leading order we find the following scaling solutions and their deforma-
tions:
f(r)e−β(r) = r2 ,
f(r) = f0r2(3−b2)
3 ,
At(r) = 0 ,
Φ(r) =2b√
3log r , (3.15)
with
f0 =3V 2
0
4(9− b2). (3.16)
These solutions have dynamical critical exponent z and hyperscaling parameter θ given
by
z = 1 , θ = − 2b2
3− b2. (3.17)
Just as in the fractionalised case, the vacuum expectation value of the charged scalar
is not fixed, δη = η0. There is then a second irrelevant deformation which introduces
the required flux. For a = b this is
δAt = aIRA r3+b2
6+νA(η0) . (3.18)
This exponent depends on the IR value of the charged scalar:
νA =3 + b2
6
(−2 +
√1 +
72q2η20
Z20f0(3 + b2)2
). (3.19)
This is merely the leading order contribution to the series solution from the deforma-
tion; there are additional subleading corrections in powers of 2νA. We thus have the
12
non-trivial consistency condition that this power be positive, which translates into the
constraint on the condensate:
η20 >
(3 + b2)2f0Z20
24q2. (3.20)
A similar structure was seen in the zero-temperature superconducting solutions of [26].
3.2 Finite Temperature
In order to complement our zero-temperature picture, and to complete our phase dia-
gram, we would like to study how these solutions extend to finite temperature. Holo-
graphically, a non-zero temperature is indicated by the presence of a (non-degenerate)
bulk horizon. We thus make the ansatz
f = f+(r − r+) + · · · ,
β = β+ + · · · ,
At = At,+(r − r+) + · · · ,
Φ = Φ+ + · · · ,
η = η+ + · · · . (3.21)
Temperature masks the singularities of the solutions of section 3.1, and the dilaton
is finite here. Despite this we will see clearly the imprint of the zero temperature
IR scaling behaviour at finite temperature. In particular, our results will exhibit the
striking influence that the quantum critical point has on the finite temperature physics.
Such solutions exhibit a superconducting instability [27, 28], whose critical temper-
ature depends on the value of the dilaton deformation [8]. This critical temperature
can be dialled to zero, forming the edge of a superconducting dome. We have checked
that the free energy of the broken phase solution is lower than that of the unbroken
phase whenever it exists.
Since we are holding the theory at finite chemical potential, it is natural to consider
the phase diagram as a function of Φ1/µ. We can illustrate the cohesive to partially
fractionalised critical behaviour at finite T by approaching T = 0 from the top of the
superconducting dome (represented by a black dashed line in each case) at constant
values of Φ1/µ. A selection of such slices is shown in Fig. 2. We see that the critical
solution approaches the unique constant dilaton solution in the IR, while solutions on
either side of it exhibit an IR divergence as T = 0 is approached, consistent with our
earlier analysis.
13
èè-2 0 2 4 6
0.00
0.02
0.04
0.06
0.08
0.10
èè-1.5 -1.0 -0.5 0.0 0.5
0.00
0.02
0.04
0.06
0.08
0.10
èè0.2 0.3 0.4 0.5 0.6 0.70.00
0.02
0.04
0.06
0.08
0.10
èè0.0 0.1 0.2 0.3 0.4 0.50.00
0.02
0.04
0.06
0.08
0.10
Tµ
Φ+Φ1
µ
Qµ2
η2
µ2
Figure 2: Cohesive to partially fractionalised critical behaviour in the brokenphase. The dashed black line indicates the edge of the superconducting domeand the coloured lines are some slices at fixed Φ1/µ near the critical solution(black). The black dot shows the critical T = 0 solution with good agreementwith the critical T > 0 solution. Note that the fixed Φ1 slices in the bottomleft panel are slightly curved, signifying a small variation of the charge withtemperature at fixed chemical potential and fixed UV deformation Φ1.
A comprehensive visualisation of the critical behaviour is given in Fig. 3 where we
show a fit to the scaling α of the entropy density of the form
s
µ2∼ c1
(T
µ
) 2α
, (3.22)
for the region below the superconducting dome. From equation 2.1 we expect that near
T = 0:
α ∼ z
1− θ2
. (3.23)
The emergence of the quantum critical wedge associated with the z = 1 quantum
critical point of the cohesive to partially fractionalised ground state is clearly visible.
On either side of the wedge, the values of the coefficient α are in agreement with
14
the predicted values of α = 2/3 in the cohesive phase and α = 2 in the (partially)
fractionalised phases.
Tµ
Φ1
µ
1.0
0.75
QCP
Figure 3: Scaling behaviour for the quantum-critical region associated with thez = 1, θ = 0 QCP. Colour indicates the value of α, which is proportional to thelogarithmic derivative of entropy with respect to temperature at fixed µ. TheT = 0 QCP is located at the black dot. Hatching indicates low temperatureregions not covered by our data.
3.3 The Fractionalisation Transition
Our analysis thus far can be taken as strong evidence for the existence of a T = 0 phase
transition between a superfluid cohesive phase and a superfluid fractionalised phase at
a critical UV deformation Φ1/µ(gm) ' −0.131. In this section we complete the physical
picture, deforming the IR geometries of section 3.1 by irrelevant deformations in order
to construct full solutions which asymptote to AdS4 in the UV.
Our numerical results are conclusive: we find one-parameter families of full ge-
ometries of each of the three types — superfluid cohesive phase, superfluid fraction-
alised phase and fully fractionalised. The superfluid cohesive phase exists for Φ1/µ <
Φ1/µ(gm). The superfluid fractionalised phase exists for Φ1/µ(gm) < Φ1/µ < Φ1/µ(gf ),
15
meeting the fully fractionalised branch smoothly at Φ1/µ(gf ) ' 0.621.
A holographic measure of fractionalisation is the ratio of flux emanating from the
deep IR of the solution (e.g. a black-hole horizon) and the total charge Q:
AQ≡ 1
Q
(√−gZ(Φ)F rt
∣∣∣r+
). (3.24)
As a consequence of Gauss’s Law, this must interpolate between zero for the superfluid
cohesive phase and unity in the fully fractionalised phase [1, 3, 21]. In Fig. 4 we plot
this measure for each of the solution branches found, which confirms their identities as
either superfluid cohesive, superfluid fractionalised or fully fractionalised.
-0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0
-0.5 0.0 0.5 1.0
-0.20
-0.15
-0.10
-0.05
0.00
AQ
ωµ3
Φ1/µΦ1/µ
gc gfgc
gf
Figure 4: Dependence of the T = 0 domain walls on the parameter Φ1/µ. Asin Fig 1, shades of red indicate broken U(1) symmetry, with dark red indicatinga superfluid cohesive and bright red a superfluid fractionalised phase. Fullyfractionalised geometries are shown in green. Supplementary low temperaturedata (T ' 10−3µ) in the vicinity of the fractionalisation transition gc is indicatedby black dashes.
For the above phase transitions to occur, it is necessary that the fully fractionalised
branch be thermodynamically subdominant wherever it coexists with the superfluid
cohesive or superfluid fractionalised branches. To this end, we plot the free energy
density ω of each family in Fig. 4. We see that the free energy is indeed higher in the
fully fractionalised phase. It proved numerically unprofitable to calculate the T = 0
free energies of the cohesive and superfluid fractionalised branches in a small region
close to the critical point, so in that region we have supplemented our results with low
temperature (T ' 10−3µ) data. Furthermore we note the precise agreement between
the T = 0 analysis and the low temperature results, which is thanks to the plateauing
of the thermodynamic quantities near T = 0 that we (numerically) observed in Fig. 2.
The behaviour of the free energy in Fig. 4 strongly suggests that the phase transition
at gc is continuous. Indeed, we have computed the first two derivatives of the free energy
16
with respect to φ1 and have found them to be continuous about gc, implying at least a
third-order transition. Our supplementary data does not allow us to study arbitrarily
high derivatives with sufficient accuracy.
In contrast, we find a kink in the first derivative of the free energy at gf , implying
a second-order transition there. This tallies well with the fact that, in our model, the
transition at gf is one between a superconducting phase and a normal phase.
Both these results can be compared to the case of fermionic fractionalisation transi-
tions [1], where the roles are reversed. The analogous transition between their ‘mesonic’
(here called ‘cohesive’) and partially fractionalised phases are first or second order, de-
pending on the parameters of the model, while the transition between partial and full
fractionalisation is continuous of third order.
3.4 Class Ib: A Bottom-Up Model
The results we have presented so far map out the detailed phenomenology for a theory
in class Ia, under the classification of Fig. 1, for which the coupling term ZF remains
finite for an interval of UV scalar deformations Φ1. Consequently, we argue, the theory
in this interval was able to support superfluid cohesive solutions which provided the
thermodynamically preferred solutions.
Here we make a few brief comments on a model governed by the same potential
(3.1), but with a Φ-even gauge coupling term ZF (Φ) = Z20 cosh
(aΦ√
3
)with a > 0. The
important feature of this model is that ZF will diverge in the IR given an IR divergent
Φ, irrespective of its sign. Thus, the interval of cohesive solutions exhibited by our
earlier model will here be reduced to a single point, Φ1 = 0. At this point the bulk
solution has Φ(r) = 0 everywhere and is simply an AdS4 to AdS4 charged domain wall.
Thus our model exhibits a transition from superfluid cohesive phase at the point
Φ1 = 0, to superfluid fractionalised phase 0 < |Φ1| < Φ1(gf ), and ultimately to a fully
fractionalised phase when |Φ1| ≥ Φ1(gf ). This is illustrated in Fig 1. We now turn to a
model which arises from a consistent truncation of 11D supergravity, which shares even
gauge coupling Z with class Ib, but which differs from all models in class I by having
finite Z everywhere. As we shall see this leads to markedly different phenomenology.
17
4 M-Theory
We work with the consistent truncation of [8, 29], whose equations of motion can be
obtained from the action
SM =1
16πG
∫d4x√−g[R− (1− h2)3/2
1 + 3h2FMNF
MN − 3
2(1− 34|χ|2)2
|Dχ|2
− 3
2(1− h2)2(∇h)2 − 6
`2
(−1 + h2 + |χ|2)
(1− 34|χ|2)2(1− h2)3/2
]+
1
16πG
∫2h(3 + h2)
1 + 3h2F ∧ F .
(4.1)
By defining χ = ξeiqϕ and performing the transformation
h = tanh
(Φ√3
), ρ =
2√3
tanh
(η√2
), (4.2)
this action falls into the class II of models of (2.2, 2.3) with the specific choices
ZF (Φ) =4
cosh3(
Φ√3
)(1 + 3 tanh2
(Φ√3
)) , X(η) =1√2
sinh(√
2η)
(4.3)
and charge q` = 1, [7]. This model also has a Chern-Simons term. We now turn to
a description of the detailed phase structure of the CFT duals of this system, held at
finite density, with particular regard to the zero-temperature ground states. Much of
this discussion has been reported in the literature [7, 8], and more details can be found
there.
4.1 Phases
As before, a superfluid branch of solutions emerges as an instability of the charged
Reissner-Nordstrom family of solutions which exists in this model when h = 0. Once
again, under the dilaton deformation in the constant µ phase diagram we see a dome of
superconducting solutions, whose T = 0 limit, as shown in [7, 8], is a charged domain
wall interpolating between two AdS regions of different AdS lengths. Whenever the
U(1) symmetry remains unbroken the zero-temperature limit of the neutral and charged
black hole solutions is incompressible, in the sense that it does not depend on the value
of µ. In fact it approaches a zero-density state with vanishing (direct) conductivity.
Furthermore, the T → 0 limit of the neutral and charged solutions meet at the unique,
unbroken T = 0 solution of the system, which lifts to an eleven-dimensional Schrodinger
18
solution of M-theory [30]. Thus the entire region6 at T = 0 outside the dome is
degenerate on a constant-µ phase diagram. In view of this ‘incompressibility’ of the
groundstate, Fig. 5 shows the domains where these solutions exist at fixed energy
rather than fixed chemical potential as in [8].
As we tune the deformation by Oh, both at T = 0 and at T 6= 0, the expectation
from our previous results in this paper is that we eventually reach a transition from
the broken U(1) solution to a (partially) fractionalised phase.
Extremal charged BH& charged domain walls
unbroken charged BHs& superfluids
unbroken chargedBHs
T
h
Schwarzschild
Schrödinger
RN
RN
& s
.c.s
Figure 5: The existence of bulk solutions at fixed mass. The red line denotesneutral solutions terminating in the Schr zero-termperature fixed point. Notethat the neutral dome meets the superconducting dome at precisely this fixedpoint, showing that there is no fractionalised phase in this model.
We now turn to an exploration of the neutral solutions of (4.1) outside the dome in
which the neutral scalar h attains the singular value h = 1 in the IR. In the parametri-
sation of Eq. (4.2) we see that this corresponds to a (logarithmically) divergent dilaton
singularity, which should by now be no surprise.
6The possible emergence of non-isotropic phases [31, 32] is beyond the scope of this work, but couldmodify the picture in some regions.
19
4.2 Neutral top-down solutions
We rewrite (4.1) for the case of interest assuming the charged scalar χ is trivial, since
we do not want solutions in which the U(1) symmetry is broken. Hence, passing to the
variables (4.2), we find
S =
∫d4x√−g[R−
sech3(
Φ√3
)1 + 3 tanh2
(Φ√3
)F 2 − 1
2(∇Φ)2 +
6
`2cosh
(Φ√3
)]. (4.4)
Recall that at finite temperature we have a one-parameter family of neutral ‘dilatonic’
black holes, with IR behaviour (3.21), where ξ ≡ 0. At zero temperature the nature of
the solution changes, as we now describe. Guided by our findings above we suppose7
that the isometries of the metric are enhanced from the R × SO(2) symmetry above
to the full SO(1, 2) Lorentz symmetry. By choosing a convenient radial gauge we take
ds2 =dρ2
F (ρ)+ ρ2sech
(Φ(ρ)/
√3)ηµνdx
µdxν . (4.5)
This expression is convenient for the current purposes, but we note that we can convert
it into the form (2.6) via
r2 = ρ2 sech(
Φ/√
3), e−βf = r2 ,
√F =
dρ
dr
√f . (4.6)
We can use this ansatz to construct the singular IR solutions admitted by the M-theory
truncation.
4.2.1 Singular IR solutions
We can construct the desired IR solution by a standard dimensional reduction trick.
Suppose then that Φ diverges8 in the IR as Φ→∞. In this limit the action approaches
Ssing =
∫d4x√−g[R− 2e−
√3ΦF 2 − 1
2(∇Φ)2 +
3
`2eΦ/√
3], (4.7)
which can be obtained in a reduction via a gravi-photon ansatz
ds2 = eΦ/√
3ds2(M) + e−2Φ/√
3(dz + 2
√2A)2
, (4.8)
7In support of this assumption we have strong numerical evidence for this behaviour by studyingthe zero-temperature approach of the none-extremal black hole solutions.
8The case Φ→ −∞ is equivalent by the Φ→ −Φ symmetry of the lagrangian coming from 11D.
20
where A ∈ T ∗M and hats denote five-dimensional quantities. The starting five-
dimensional action is simply Einstein-Hilbert with a cosmological constant:
S5 =
∫d5x√−g[R +
3
`2
]. (4.9)
Under this reduction, pure five-dimensional AdS with metric
ds25 = `2
5
[dρ2
ρ2+ ρ2
(−dt2 + dx2
)+ ρ2dz2
],
(`2
5 = 4`2), (4.10)
reduces to a logarithmic dilaton solution in four dimensions, that is:
ds2 = `25
[dρ2
ρ+ ρ3(−dt2 + dx2)
], with Φ = −
√3 log(ρ) . (4.11)
This solution, as we shall now see, plays the role of the singular IR we are after. Note
that in this case the gravi-photon field A is trivial, so that the IR solution is obtained
via a simple circle reduction from five dimensions.
Note that we also have a five-dimensional Schrodinger metric [33, 34, 35], obtained
as a simple deformation of the above, by adding a term proportional to the light-cone
coordinate (dx+)2:
ds2 = `25
[−β2ρ4(dx+)2 +
dρ2
ρ2+ ρ2
(dx2
2 + 2dx+dx−)]
, (4.12)
where β is a constant. Once reduced to four dimensions and seen as an expansion
in small ρ, the Schrodinger and AdS metrics in fact agree to leading order, that is
they give the same IR behaviour for the desired logarithmic dilaton solution in four
dimensions. Of course, solving the full equations coming from (4.1) order by order will
determine the exact solution and thus distinguish between the two cases. This was
pointed out earlier in [30].
Here we demonstrate that this solution is in fact the universal T → 0 limit of the
model (4.1) outside the dome. The full M-theory solution, lifted to eleven dimensions,
is a Schrodinger solution Schr5 ×KE6 with dynamical critical exponent z = 2, where
the Schr5 is fibred9 non-trivially over the Kahler-Einstein space KE6. In the dual
field theory this solution corresponds to a mass-deformation driving the system to the
non-relativistic z = 2 IR fixed point [30]. The full structure of this is not evident
from the four-dimensional perspective, but we see the z = 2 scaling for example in the
temperature scaling of entropy density at criticality.
9Similar solutions were constructed previously in [36].
21
Returning thus to constructing the neutral IR solutions, we substitute the metric
ansatz into the equations following from (4.1), resulting in a single, fully decoupled,
non-linear ODE for the scalar field Φ. Its precise form is given in appendix A. Once Φ
is determined, the Einstein equations determine the metric function F (ρ) algebraically:
F (ρ) =4`−2ρ2 cosh
(Φ/√
3)
4− (ρΦ′/√
3)2sech2(Φ/√
3)− 4ρΦ′/
√3 tanh
(Φ/√
3) , (4.13)
where the prime denotes a derivative with respect to the radial direction ρ. The Φ
equation admits an IR expansion, which reproduces (4.11) to leading order. It takes
the form
Φ(ρ) = −√
3 log ρ+3√
3
2ρ2 + · · · , (4.14)
and contains no free parameters. Note that the freedom associated with the ρ rescaling
of the metric has no physical effect on the solution. This unique IR expansion can be
integrated to the UV and connects to the desired skew-whiffed AdS4 solution. We
have done this and read off the UV data. In appropriate units10 one finds h2/h21 =
−0.288 . . . ∼ −1/(2√
3) for the expectation value 〈Oh〉 and ε/h31 = 0.779 . . . ∼ 3
3√
3
for the energy. The analytic values are those of the exact solution found in [8]. One
can check that the chemical potential µ(r∞)− µ(rIR) vanishes. We thus conclude that
the unique neutral T = 0 solution of the theory (4.1) with logarithmically diverging
dilaton is the analytic solution given in Eq (8.2) of [8].
4.3 Approaching the critical point
It is enlightening to study the power-law associated with the approach to zero tem-
perature. At the critical value we see scaling behaviour compatible with the z = 2
fixed point, in other words T -linear scaling. Consulting Fig. 6 we see that this scaling
behaviour is indeed true precisely, if the temperature is lowered holding Φ1/µ at the
critical value. The critical curve in black clearly approaches α = 2 in the appropriate
limit. The red curve corresponds to the approach of the (unstable) AdS2 infrared, and
so the diverging z → ∞ is expected. Note that this region is cloaked by the super-
conducting dome in the full phase diagram. The blue curve corresponds to a value
of h1/µ outside the dome. The value of α remains bounded as T approaches zero in
this case. While the zero-temperature solution itself, for h1/µ > h1c/µ is independent
of the value of the chemical potential the finite-temperature solutions are not, and so
10We quote numerical values in the same convention as [8] for ease of comparison.
22
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1
2
3
4
5
T/µ
α
Figure 6: Approaching the M-theory Schrodinger critical point from finitetemperature. The black line for h1/µ = h1c/µ = 0.354... shows precisely thecritical T -linear scaling. It delineates the blue supercritical line (h1/µ = 0.382)from the red subcritical line (h1/µ = 0.324). Note that the complex scalar fieldis set to zero in all cases here, and the subcritical region is masked in the fulltheory by a superconducting dome.
the approach does depend on ratios like h1/µ or T/µ. This is clearly illustrated in
Fig. 6, where the critical scaling with α = 2 delineates a region of supercritical (as
exemplified by the blue curve) scaling with α < 2 from a region of subcritical scaling
with α → ∞, as exemplified by the red curve. The latter behaviour corresponds to
approaching the zero-temperature AdS2 geometry, which in the full theory is masked
by the superconducting dome.
5 Discussion
The results of this paper are twofold. On the one hand we have constructed bosonic
analogues of so-called ‘fractionalisation transitions’ by relying on suitably constructed
minimal bottom-up models. On the other hand we reexamined the phase diagram of
the M-theory superconductor in this light, showing how the analogous transition in
this system is in fact a transition from a superfluid cohesive phase to a neutral phase
with vanishing (direct) conductivity11. We emphasise again, that these transitions are
taking place at constant chemical potential.
11Interestingly, however, there is a non-vanishing transverse conductivity Reσxy 6= 0.
23
The IR behaviour of the dilaton coupling to the gauge kinetic term at zero tem-
perature governs the quantum phase transitions observed in this paper. All examples
encountered in this work support the conjecture that one cannot have a cohesive phase
if the dilaton coupling to the gauge-kinetic term diverges (Z →∞). While we do not
have a rigorous argument that would prove this, we can see heuristically that it makes
sense: diverging Z means that the effective gauge coupling vanishes as is obvious in a
normalisation where the gauge-kinetic terms reads ∼ 1e2F 2. That means that matter in
the strict IR, where Z diverges, cannot source any flux. Thus any flux there can only
emanate from a horizon and consequently we would have a fractionalised contribution
to the overall flux. Our arguments for this behaviour are purely holographic, that is
we use the dynamics of the bulk gravity. Similarly our classification of the different
phases are based on such bulk arguments. Clearly it would be very interesting to use
such bulk arguments to shed more light on the nature of a possible order parameter
for the fractionalisation transitions of the kind exemplified in this paper.
In asymptotically AdS spacetimes, one often finds that the zero temperature limit
of a black hole solution has an emergent AdS2 geometry, indicated by a divergent
dynamical critical exponent. The dual field theory then has a non-zero entropy at
zero temperature. Often these finite entropy geometries are unstable when embedded
in top-down models (see e.g. [7, 8, 31]), so that the third law of thermodynamics is
upheld by the classical geometry.
As is clear from equation (2.1), and strikingly visualised in Fig. 3, the T = 0
infared hyperscaling parameter governs the scaling behaviour of various thermodynamic
quantities at low temperature. In particular, a divergent critical exponent can be
compensated for by an equally divergent hyperscaling parameter [11], resulting in a
vanishing entropy at T = 0. In this manner one can avoid instabilities associated
with finite entropy. Our bottom-up model of section 3 uses precisely this mechanism.
The M-theory model, however, is of class II, and so there is no fractionalised phase
associated with a vanishing gauge coupling and its corresponding hyperscaling violation
solutions; we indeed see that the fractionalised (Reisner-Nordstrom) phase is unstable
to scalar condensation, and so masked by a superconducting dome. It would be very
interesting to investigate precisely what conditions have to be met for any entropic
singularity (i.e. any AdS2 IR geometry) to be unstable [37] to new phases cloaking it
at low temperatures.
Recently, following earlier work on Lifshitz geometries [38], it has been suggested
[39, 40] that hyperscaling IR geometries are destabilised due to the running dilaton.
24
If this is the case for the geometries considered here then our results will still apply
for an intermediate range of energy scales, above the scale where quantum corrections
smooth out the IR geometry. It is interesting in this context that at least some of
these seemingly singular geometries can be lifted to perfectly regular higher-dimensional
geometries (see also [22]), potentially eliminating the need to consider the corrections of
[39, 40]. Note, however, that in the case of Lifshitz geometries, the higher-dimensional
lifts were found to be as problematic as their lower-dimensional descendants [41], as
manifested in the propagation of test strings. The status of the possible resolution
of such singularities (e.g. by matter sources [42]) in string theory deserves further
attention.
Acknowledgements
It is a pleasure to thank Aristomenis Donos, Jerome Gauntlett, Andrew Green, Sean
Hartnoll, David Tong and Subir Sachdev for discussions. JS would like to thank the
MCTP at the University of Michigan and the CTP at the Massachusetts Institute of
Technology for hospitality while this work was in progress. AA and BC are supported
by STFC studentships. BW is supported by the Royal Commission for the Exhibition
of 1851.
A The ODE in section 4.2.1
Here we give the details of the ODE determining Φ(ρ) in section 4.2.1, which follows
from the lagrangian (4.7) upon setting At = µ with µ constant.
ρ2Φ′′(ρ) =−1
2√
3tanh
(Φ(ρ)√
3
) [12− ρ2Φ′(ρ)2
(4 + 3 sech2
(Φ(ρ)√
3
))]−[1 + 3 sech2
(Φ(ρ)√
3
)]ρΦ′(ρ) +
1
12sech4
(Φ(ρ)√
3
)ρ3Φ′3(ρ) . (A.1)
For any solution the chemical potential in the boundary theory is arbitrary, as A is
pure gauge in the bulk. If A is to be manifestly well-defined in Kruskal coordinates
one should choose the gauge At = 0.
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